Of course he is

But only socially I'm assuming. Its obvious in a lot of his interviews. You see it most when the person he is interviewing has a stronger personality or starts taking over the interview. He just agrees with them and laughs at all their jokes, sometimes before even hearing the end of what they're actually saying. Or hell rush threw what he's saying and skip over words. He also stutters and displays nervous body language as well. Why else would he had/have a drinking problem if he seems like such a happy guy all the time and is so successful, bc alcohol calms your nerves and makes it easier to be social. He might be highly sensitive as well or just has social anxiety. Either way i commemorate him for taking his career to where it is today. I can only imagine what its like doing show business while having to constantly subdue your anxiety.

Being nervous is part of the platform

When you take over a network television show like the tonight show anyone would be nervous doing it for the first time. Jimmy Fallon is a great television personality but still has to get used to a stage and show of the magnitude on national television. After a couple of shows he will be comfortable and should be better.

Based on his past behavior on SNL, probably

Jimmy Fallon had a bad habit of breaking character during skits on SNL, because he couldn't contain his laughter. I think the fact that he had such a hard time controlling his laughter is a sign overall that he tends to have trouble containing his emotions while he's performing, and that it's not really an act that we're seeing but him being himself. In fact, there's a fair chance that the reason why he broke down so much during his SNL was to cover up for his anxiety.

He is sometimes.

I will say that Jimmy Fallon does come off a bit as the shy, nerdy, awkward type. I think that he does often seem genuinely nervous when he's on TV. I guarantee he was very nervous a few nights ago when he debuted as the new host of The Tonight Show. He had some big shoes to fill.

No, I believe it is an act.

No, I do not believe the nervousness is genuine. As a comedic actor/host, Jimmy Fallon is used to doing what is necessary for a laugh, and has found a lot of success in his nervous character. Flubbed lines, glancing at the camera, and other signs of nervousness have become common in Jimmy's on-screen personas.

Jimmy Fallon is a character.

Jimmy Fallon has been entertaining for a long time. His nervousness is now an act. He has certain reactions to certain things. He scripts this. His delivery is excellent and his job as an entertainer is fulfilled. If you think he is genuinely nervous, then he has done an excellent job at entertaining you.